MATH340, Fall 2015
EXAM I
Colorado State University
EXAM I
5:00-7:00PM, Oct 8, 2015
Name:
Section:
• You have two hours to complete this exam.
• No notes, books, or other references are allowed during this exam.
• Calculators are not allowed during the exam.
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• A two-sided cheat sheet of 8 21 × 11 or smaller is allowed during the exam.
• There are questions on the front of the page. You can use the back of each page if you
need more space. If you do use any extra paper, make sure that your name is on it and
that you attach it to your exam.
• You must show all work to receive credit. Answers for which no work is shown will
receive no credit unless specifically stated otherwise.
Question Score Maximum
1
20
2
15
3
15
4
15
5
20
6
15
Total
100
1
MATH340, Fall 2015
EXAM I
Colorado State University
1. (20 points) True (T) or False (F) problems (circle T or F only, no partial credit):
(i) T or F
If x0 = x cos2 t and x(0) = 1, then x(t) > 0 for all t for which x is defined.
(ii) T or F
The initial value problem y 0 = y 1/4 , y(0) = 3 has a unique solution.
(iii) T or F
The differential equation sin(x + y)dx + (2y + sin(x + y))dy = 0 is not exact.
(iv) T or F
The equation (xy − 2)dx + (x2 − xy)dy = 0 can have the function µ(x) = 1/x2
as its integrating factor.
(v) T or F
The equation x0 = (x − t)(x + t) is not autonomous.
(vi) T or F
The three equilibrium points for the equation y 0 = f (y) in the figure below
are asymptotically stable (A), unstable (B), and asymptotically stable (C), respectively.
f(y)
B
A
y
C
(vii) T or F
The null space of the 1 × 3 matrix A = (−2, 3, 1) is a one-dimensional space.
(viii) T or F
A nontrivial subspace V (i.e., V 6= {0}) of Rn has a unique basis.
(ix) T or F
The solution curve of an ODE is tangent to the direction field of the equation.
(x) T or F
An ODE is exact if and only if it is separable.
2
MATH340, Fall 2015
EXAM I
Colorado State University
2. (15 points) Consider a linear ordinary differential equation (2t + 3)x0 = x + (2t + 3)1/2
for function x(t).
(i) Find the general solution;
(ii) Find the particular solution satisfying x(−1) = 0.
3
MATH340, Fall 2015
EXAM I
Colorado State University
3. (15 points) Consider a linear system Ax = b given by
x1 − x2 − 2x3 = 1
2x1 + x2 − x3 = 2
2x1
− 2x3 = 2.
(i) Apply elementary row operations to simplify the augmented matrix to the reduced row
echelon form;
(ii) Write down the solutions of the linear system in a parametric form;
(iii) Find the dimension and a set of basis vectors of the nullspace of the matrix A.
4
MATH340, Fall 2015
EXAM I
Colorado State University
4. (15 points) Consider an ordinary differential equation tx0 = 2x − t, x(0) = 2.
(i) Find the general solution of the equation. Sketch several members of the family of
solutions portrayed by the general solution;
(ii) State an existence theorem for the solution of ODE.
(iii) Show that there is no solution satisfying the given initial condition. Explain why this
lack of solution does not contradict the existence theorem.
5
MATH340, Fall 2015
EXAM I
Colorado State University
5. (20 points) Consider an autonomous differential equation x0 (t) = f (x) where f (x) =
(x + 1)(x2 − 9).
(i) Sketch a graph of function f (x);
(ii) Use the graph of function f (x) to develop a phase line for the autonomous equation.
Classify each equilibrium point as either unstable or asymptotically stable;
(iii) Sketch the equilibrium solutions in the tx-plane. These equilibrium solutions divide the
tx-plane into regions. Sketch at least one solution trajectory in each of these regions.
6
MATH340, Fall 2015
EXAM I
Colorado State University
6. (15 points) Suppose that (xy − 1)dx + (x2 − xy)dy = 0 has an integrating factor that is
a function of x alone. Find the integrating factor and use it to solve this differential equation.
7